A construction of Shatz strata in the polystable $G_2$-bundles moduli space using Hecke curves

Let $X$ be a compact Riemann surface of genus $g\ge 2$ and $M(G_2)$ be the moduli space of polystable principal $G_2$-bundles over $X$. The Harder-Narasimhan types of the bundles induced a stratification of the moduli space $M(G_2)$ called Shatz stratification. In this paper, a description of the Shatz strata of the unstable locus of $M(G_2)$ corresponding to certain family of Harder-Narasimhan types (specifically, those of the form $(\lambda ,\mu ,0,-\mu ,-\lambda )$ with $\mu <\lambda \le 0$) was given. For this purpose, a family of vector bundles was constructed in which a 3-form and a 2-form were defined so that it was proved that they were strictly polystable principal $G_2$-bundles. From this, it was proved that, when the genus of $X$ was $g\ge 12$, these Shatz strata were the disjoint union of a family of $G_2$-Hecke curves in $M(G_2)$ that will be constructed along the paper. Therefore, the presented results provided an advance in the knowledge of the geometry of $M(G_2)$ through the study of its Shatz strata and presented a methodological innovation, by using Hecke curves for this study.